📚 Year 7 OCR Maths: Core Knowledge Points Review | Year 7 OCR 数学：核心知识点梳理
In Year 7, students following the OCR curriculum build a strong foundation across number, algebra, geometry, and statistics. Mastering these core topics early makes later years much smoother. This article brings together all the essential knowledge points you need, explained clearly with paired English and Chinese explanations, so that every concept clicks.
在七年级，学习 OCR 课程的学生将在数、代数、几何和统计方面打下坚实的基础。尽早掌握这些核心主题会让后续的学习顺畅许多。本文汇集了你需要掌握的所有基本知识点，并以中英文对照的方式清晰讲解，让每一个概念都了然于心。
1. Place Value and Rounding | 位值与四舍五入
Place value tells us how much each digit in a number is worth, depending on its position. For example, in the number 4,739 the digit 4 is in the thousands place, so it stands for 4,000; the 7 is in the hundreds place (700); the 3 is in the tens place (30); and the 9 is in the units or ones place (9).
位值告诉我们一个数字中每个数位上的数字值多少。例如，在数字 4,739 中，数字 4 位于千位，代表 4,000；7 位于百位（700）；3 位于十位（30）；9 位于个位（9）。
Rounding simplifies a number while keeping it close to the original value. If the digit to the right of the rounding place is 5 or more, we round up; if it is 4 or less, we round down. For example, 2,648 rounded to the nearest hundred is 2,600 because the tens digit is 4.
四舍五入能让数字简化，同时保持接近原值。如果要舍入的位置右边一位数字是 5 或更大，就“入”；如果是 4 或更小，就“舍”。例如，2,648 四舍五入到百位是 2,600，因为十位数字是 4。
253 rounded to the nearest 10 → 250; 376 rounded to the nearest 100 → 400
253 四舍五入到十位 → 250；376 四舍五入到百位 → 400
2. Operations with Whole Numbers | 整数的运算
The four operations are addition, subtraction, multiplication, and division. For column addition and subtraction, always align digits by place value. Multiplication and division can be performed using written methods or mental strategies, such as breaking numbers into smaller parts.
四则运算是加法、减法、乘法和除法。进行竖式加减法时，一定要将相同数位对齐。乘除运算可以使用竖式或心算策略，例如将数字拆分成较小的部分。
Order of operations is remembered by BIDMAS or BODMAS: Brackets first, then Indices (or Orders, i.e. powers), then Division and Multiplication (from left to right), and finally Addition and Subtraction (from left to right). For instance, 3 + 4 × 2 = 3 + 8 = 11, not 14.
运算顺序可以用 BIDMAS 或 BODMAS 记忆：先算括号 (Brackets)，接着算指数 (Indices/Orders，即乘方)，然后算乘除 (从左到右)，最后算加减 (从左到右)。例如，3 + 4 × 2 = 3 + 8 = 11，而不是 14。
3. Factors, Multiples, and Primes | 因数、倍数与质数
A factor is a whole number that divides exactly into another number with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. A multiple is the product of a number and any whole number, so the first few multiples of 5 are 5, 10, 15, 20, …
因数是能整除另一个数而没有余数的整数。例如，12 的因数有 1、2、3、4、6 和 12。倍数则是一个数与任意整数的乘积，所以 5 的前几个倍数是 5、10、15、20……
A prime number has exactly two distinct factors: 1 and itself. The first five primes are 2, 3, 5, 7, and 11. The highest common factor (HCF) of two numbers is the largest factor they share, while the lowest common multiple (LCM) is the smallest multiple they share.
质数恰好有两个不同的因数：1 和它本身。前五个质数是 2、3、5、7 和 11。两个数的最高公因数 (HCF) 是它们共有的最大因数，而最小公倍数 (LCM) 是它们共有的最小倍数。
4. Introduction to Negative Numbers | 负数入门
Negative numbers are numbers less than zero. On a number line they lie to the left of zero. As you move left, the numbers get smaller: -5 is less than -2.
负数是小于零的数。在数轴上它们位于零的左边。向左移动时，数字变小：-5 比 -2 小。
Adding a negative number is the same as subtracting its positive counterpart: 6 + (-3) = 6 – 3 = 3. Subtracting a negative number is the same as adding the positive number: 4 – (-2) = 4 + 2 = 6. When you have two signs side by side, ‘same signs’ make a plus and ‘different signs’ make a minus.
加一个负数等于减去它的相反数：6 + (-3) = 6 − 3 = 3。减去一个负数等于加上它的正数：4 − (−2) = 4 + 2 = 6。当两个符号碰在一起时，“同号”得加号，“异号”得减号。
5. Fractions, Decimals, and Percentages | 分数、小数和百分数
Fractions, decimals, and percentages are three ways of showing parts of a whole. To change a fraction to a decimal, divide the numerator by the denominator. To change a decimal to a percentage, multiply by 100. For example, 3/5 = 0.6 = 60%.
分数、小数和百分数是表示整体的一部分的三种方式。把分数化成小数，用分子除以分母。把小数化成百分数，乘以 100。例如，3/5 = 0.6 = 60%。
When adding or subtracting fractions, the denominators must be the same. For multiplication, multiply the numerators together and the denominators together. To find a percentage of an amount, write the percentage as a decimal or fraction and then multiply: 30% of 200 = 0.3 × 200 = 60.
加减分数时，分母必须相同。乘法时，分子相乘作分子，分母相乘作分母。求一个数量的百分之几时，先把百分数写成小数或分数再乘：200 的 30% = 0.3 × 200 = 60。
1/2 = 0.5 = 50%; 1/4 = 0.25 = 25%; 3/4 = 0.75 = 75%
1/2 = 0.5 = 50%；1/4 = 0.25 = 25%；3/4 = 0.75 = 75%
6. Sequences and the nth Term | 数列与通项
A sequence is an ordered list of numbers that follows a rule. In a linear (arithmetic) sequence the difference between consecutive terms is constant. For example, in 4, 7, 10, 13, … the term-to-term rule is ‘add 3’.
数列是一组按规则排列的数。在线性（等差）数列中，相邻两项的差是常数。例如，4, 7, 10, 13, … 的递推规则是“加 3”。
The position-to-term rule (nth term) lets you find any term without knowing the previous one. For the sequence 3, 5, 7, 9, … the nth term is 2n + 1. So the 10th term is 2×10 + 1 = 21.
通项公式（第 n 项）使你不需要知道前一项就能求出任意项。对于数列 3, 5, 7, 9, …，通项是 2n + 1。所以第 10 项是 2×10 + 1 = 21。
7. Algebraic Expressions and Substitution | 代数表达式与代入
In algebra, letters like x, y, or n stand for unknown numbers. An expression is a combination of letters, numbers, and operations, such as 3a + 2 or 5x − 7y. We simplify expressions by collecting like terms: 4a + 2a = 6a and 3b + 5b − b = 7b.
在代数中，x、y 或 n 这样的字母代表未知数。表达式是字母、数字和运算的组合，例如 3a + 2 或 5x − 7y。我们通过合并同类项来化简：4a + 2a = 6a，3b + 5b − b = 7b。
Substitution means replacing a letter with a given value. If a = 3 and b = 4, then 2a + b = 2×3 + 4 = 6 + 4 = 10. Always use brackets when substituting negative numbers to avoid sign mistakes.
代入是指用给定的数值替换字母。如果 a = 3，b = 4，那么 2a + b = 2×3 + 4 = 6 + 4 = 10。代入负数时永远使用括号，避免符号错误。
8. Solving One‑Step Equations | 解一步方程
An equation says that two expressions are equal, like x + 5 = 12. To solve it, we find the value of the letter that makes the equation true. The key is to do the same operation to both sides.
方程表明两个表达式相等，如 x + 5 = 12。解方程就是求出能使方程成立的字母的值。关键是对等式两边进行相同的运算。
For x + 5 = 12, subtract 5 from both sides → x = 7. For 3x = 24, divide both sides by 3 → x = 8. For x ÷ 4 = 6, multiply both sides by 4 → x = 24. Always check your answer by substituting it back into the original equation.
x + 5 = 12，两边减 5 → x = 7。3x = 24，两边除以 3 → x = 8。x ÷ 4 = 6，两边乘 4 → x = 24。永远要把答案代回原方程检验。
9. Angle Rules and Lines | 角的法则与直线
Acute angles are less than 90°, right angles are exactly 90°, obtuse angles are between 90° and 180°, and reflex angles are greater than 180°. Angles on a straight line add up to 180°.
锐角小于 90°，直角等于 90°，钝角介于 90° 和 180° 之间，反射角大于 180°。一条直线上的角相加等于 180°。
Angles around a point sum to 360°. Vertically opposite angles are equal. In any triangle the three interior angles always sum to 180°. When a straight line crosses two parallel lines, corresponding angles are equal and alternate angles are equal.
绕一点的角之和为 360°。对顶角相等。在任意三角形中，三个内角之和总是 180°。当一条直线与两条平行线相交时，同位角相等、内错角相等。
a + b + c = 180° (triangle) ; Angles on a line → x + 50° = 180° ⇒ x = 130°
a + b + c = 180°（三角形）；直线上的角 → x + 50° = 180° ⇒ x = 130°
10. Perimeter, Area, and Volume | 周长、面积和体积
Perimeter is the total distance around the outside of a shape. For a rectangle, P = 2l + 2w or 2(l + w). Area measures the space inside a 2D shape. The area of a rectangle is length × width, while the area of a triangle is (base × height) ÷ 2.
周长是图形外部轮廓的总长。长方形的周长 P = 2l + 2w 或 2(l + w)。面积衡量二维图形内部的大小。长方形的面积 = 长 × 宽，三角形的面积 = (底 × 高) ÷ 2。
Volume measures the space inside a 3D solid. The volume of a cube or cuboid is length × width × height. Remember to state units: perimeter in cm, m; area in cm², m²; volume in cm³, m³.
体积衡量三维立体内部的空间。立方体或长方体的体积 = 长 × 宽 × 高。记住要写明单位：周长用 cm、m；面积用 cm²、m²；体积用 cm³、m³。
Area of parallelogram = base × perpendicular height
平行四边形的面积 = 底 × 垂直高
11. Coordinates and Straight‑Line Graphs | 坐标与直线图
Coordinates are written as (x, y) where x is the horizontal position and y is the vertical position from the origin (0, 0). Always move along the corridor (x) and then up or down the stairs (y).
坐标写作 (x, y)，其中 x 是从原点 (0, 0) 出发的水平位置，y 是垂直位置。永远先沿“走廊”(x) 走，再上下“楼梯”(y)。
Plotting points helps create graphs. For simple straight‑line graphs like y = 2x or y = x + 1, choose x‑values, work out the matching y‑values, and join the points with a straight line. All points on the line satisfy the equation.
描点可以画出图形。对于 y = 2x 或 y = x + 1 这样的简单直线图，选择几个 x 值，算出对应的 y 值，然后把点连成一条直线。线上的所有点都满足该方程。
12. Statistics: Averages and Charts | 统计：平均数与图表
The three common averages are: mean (sum of values ÷ number of values), median (middle value when ordered), and mode (most frequent value). The range is the difference between the largest and smallest value and measures spread.
三种常用的平均数是：平均数（总和 ÷ 数据个数）、中位数（排序后中间的值）和众数（出现最频繁的值）。极差是最大值与最小值的差，用于衡量分散程度。
Data can be displayed in bar charts for discrete data, line graphs for trends over time, and pie charts for showing proportions as angles of a circle (360° = 100%). Always label axes and give a title.
数据可以用条形图展示离散数据，用折线图展示随时间变化的趋势，用饼图通过圆心角（360° = 100%）展示比例。永远要给坐标轴加标签并写上标题。
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